DOCSLIB.ORG

A Multilevel Iteration Method for Solving a Coupled Integral Equation Model in Image Restoration

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Multilevel Iteration Method for Solving a Coupled Integral Equation Model in Image Restoration mathematics Article A Multilevel Iteration Method for Solving a Coupled Integral Equation Model in Image Restoration Hongqi Yang 1,2 and Bing Zhou 1,2,∗ 1 School of Data and Computer Science, Sun Yat-sen University, Guangzhou 510006, China; [email protected] 2 Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou 510275, China * Correspondence: [email protected] Received: 9 February 2020; Accepted: 1 March 2020; Published: 4 March 2020 Abstract: The problem of out-of-focus image restoration can be modeled as an ill-posed integral equation, which can be regularized as a second kind of equation using the Tikhonov method. The multiscale collocation method with the compression strategy has already been developed to discretize this well-posed equation. However, the integral computation and solution of the large multiscale collocation integral equation are two time-consuming processes. To overcome these difficulties, we propose a fully discrete multiscale collocation method using an integral approximation strategy to compute the integral, which efficiently converts the integral operation to the matrix operation and reduces costs. In addition, we also propose a multilevel iteration method (MIM) to solve the fully discrete integral equation obtained from the integral approximation strategy. Herein, the stopping criterion and the computation complexity that correspond to the MIM are shown. Furthermore, a posteriori parameter choice strategy is developed for this method, and the final convergence order is evaluated. We present three numerical experiments to display the performance and computation efficiency of our proposed methods. Keywords: multiscale collocation method; integral approximation strategy; multilevel iteration method; image restoration 1. Introduction Continuous integral equations are often used to model certain practical problems in image processing. However, the corresponding discrete models are often used instead of continuous models because discrete models are much easier and more convenient to implement than continuous integral models. Discrete models are piecewise constant approximations of integral equation models, and they introduce a bottleneck model error that cannot be addressed by any image processing method. To overcome the accuracy deficiency of conventional discrete reconstruction models, we use continuous models directly to restore images, which are more in line with physical laws. This idea first appeared in [1], and was widely used later, such as in [2–4]. In addition to making more sense in physics, continuous models can be discretized with higher-order accuracy. This means that the model error will be significantly decreased when compared with piecewise constant discretization, especially in the field of image enlargement. Many researchers have made great contributions to the solution of integral equations, however, many difficulties remain. First, integral operators are compact in Banach space since integral kernels are normally smooth. This will produce a situation in which the solutions of the relevant integral equations do not depend on the known data continuously. To overcome this problem, the Tikhonov [5] and the Lavrentiev [6] regularization methods were proposed to regularize the ill-posed integral equation Mathematics 2020, 8, 346; doi:10.3390/math8030346 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 346 2 of 22 into a second-kind integral equation. Second, the equation using the Tikhonov regularization method contains a composition integral operator, which greatly increases the computation time. Given this condition, a coupled system equation that only involves a one-time integral operator was proposed in [7] to reduce the high computational cost. The original second-kind integral equation involves the composition of an integral operator, which is time-consuming. The collocation method [8] and the Galerkin method [9] were proposed to discretize the coupled system, and the collocation method is much easier. The third problem is that after the coupled system equation is discretized by the collocation method, a full coefficient matrix is generated. To overcome this issue, Chen et al. [10] proposed that the integral operator be represented using a multiscale basis and obtained a sparse coefficient matrix. Then, a matrix compression technique [1,11] is used to approximate that matrix, which does not affect the existing convergence order. Finally, appropriately choosing the regularization parameter (see [12–17]) is a crucial process that should balance between the approximation accuracy and the well-posedness. The purpose of this paper is to efficiently solve the second-kind integral equation on the basis of the previous achievements of other researchers. Although the equivalent coupled system is developed to reduce the computational complexity caused by the composition of the original integral operator, this is insufficient because a lot of computing time is required to compute the integral. Therefore, inspired by [18], we further propose a fully discrete multiscale collocation method using an integral approximation strategy to compute the integration. The idea of this strategy is the use of the Gaussian quadrature formula to efficiently compute the sparse coefficient matrix. By using the piecewise Gauss-Legendre quadrature, we turn the calculation of the integration into the matrix operation, which will tremendously reduce the computation time. Another challenging issue is that directly solving the large, fully discrete system obtained from the matrix compression strategy is time-consuming. Inspired by [19], we propose a multilevel iteration method (MIM) to solve the large, fully discrete coupled system, and we further present the computation complexity of the MIM. We also propose a stopping criterion of this iteration process, and we prove that this criterion can maintain the existing convergence rate. Finally, we adopt an a posteriori choice of the regularization parameter related to the MIM and then show that it will lead to an optimal convergence order. This paper is organized into six sections. In Section2, an overview flowchart is displayed first, and then the integral equation model of the first kind is reviewed to reconstruct an out-of-focus image. Following this, an equivalent coupled system is deduced. In Section3, we present the fast multiscale collocation method to discretize the coupled system using piecewise polynomial spaces. Additionally, a compression strategy is developed to generate a sparse coefficient matrix in order to make the coupled equation easily solvable. Finally, we propose an integral approximation strategy to compute the nonzero entries of the compressed coefficient matrix, which turns the operation of the integral into a matrix computation. We also provide a convergence analysis of the proposed method. In Section4, we propose a multilevel iteration method corresponding to the multiscale method to solve the integral equations, and a complete analysis of the convergence rate of the corresponding approximate solution is shown. A posteriori choice of the regularization parameter, which is related to the multilevel iteration method, is presented in Section5, and we further prove that the MIM, combined with this posteriori parameter choice, makes our solution optimal. In Section6, we report three comparative tests to verify the efficiency of our two proposed methods. The first test shows the computing efficiency of the coefficient matrix using the integral approximation strategy and the numerical quadrature scheme in [1]. The other two tests exhibit the performance of the MIM compared with the Gaussian elimination method and the multilevel augmentation method, respectively. These tests reveal the high efficiency of our proposed methods. Mathematics 2020, 8, 346 3 of 22 2. The Integral Equation Model for Image Restoration In this section, we first depict the overall process of reconstructing out-of-focus images and then describe some common notations that are used throughout the paper. Finally, in the second subsection, we introduce the approach to formulating our image restoration problem into an integral equation. 2.1. System Overview We present an overview flowchart for reconstructing an out-of-focus image in Figure1. This process includes four main parts, that is, modeling, discretization, the parameter choice rule, and solving the equation. For the input out-of-focus image, we formulate it into a Tikhonov-regularized integral equation, which is described in the next subsection. Solving this integral equation necessitates discretization. We propose a fully discrete multiscale collocation method based on an integral approximation strategy. This method works by converting the calculation of the integral into a matrix operation. The next two parts describe the parameter choice rule and solution of the problem. We note that these two parts are actually a whole when executed in practice. However, we describe it in two parts because it is too complicated to describe as a whole. For a clearer presentation, we first display the multilevel iteration method in Section4 under the condition that we have already selected a good regularization parameter. Then, we describe how this regularization parameter is chosen in Section5. Figure 1. Overview flowchart. Some notations are needed. Suppose that Rd represents d-dimensional Euclidean space. In addition, W ⊂ R2 and E ⊂ R denote the subsets of Rd. L¥ is a special kind of space of Lp when p
Load more

Recommended publications
  • Some Schemata for Applications of the Integral Transforms of Mathematical Physics
    mathematics Review Some Schemata for Applications of the Integral Transforms of Mathematical Physics Yuri Luchko Department of Mathematics, Physics, and Chemistry, Beuth University of Applied Sciences Berlin, Luxemburger Str. 10, 13353 Berlin, Germany; [email protected] Received: 18 January 2019; Accepted: 5 March 2019; Published: 12 March 2019 Abstract: In this survey article, some schemata for applications of the integral transforms of mathematical physics are presented. First, integral transforms of mathematical physics are defined by using the notions of the inverse transforms and generating operators. The convolutions and generating operators of the integral transforms of mathematical physics are closely connected with the integral, differential, and integro-differential equations that can be solved by means of the corresponding integral transforms. Another important technique for applications of the integral transforms is the Mikusinski-type operational calculi that are also discussed in the article. The general schemata for applications of the integral transforms of mathematical physics are illustrated on an example of the Laplace integral transform. Finally, the Mellin integral transform and its basic properties and applications are briefly discussed. Keywords: integral transforms; Laplace integral transform; transmutation operator; generating operator; integral equations; differential equations; operational calculus of Mikusinski type; Mellin integral transform MSC: 45-02; 33C60; 44A10; 44A15; 44A20; 44A45; 45A05; 45E10; 45J05 1. Introduction In this survey article, we discuss some schemata for applications of the integral transforms of mathematical physics to differential, integral, and integro-differential equations, and in the theory of special functions. The literature devoted to this subject is huge and includes many books and reams of papers.
    [Show full text]
  • Integral Equation for the Number of Integer Points in a Circle
    ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS { N. 41{2019 (522{525) 522 INTEGRAL EQUATION FOR THE NUMBER OF INTEGER POINTS IN A CIRCLE Magomed A. Chakhkiev Gagik S. Sulyan Manya A. Ziroyan Department of Applied Mathematics Social State University of Russia Wilhelm Pieck St. 4, Moscow Russian Federation Nikolay P. Tretyakov Department of Applied Mathematics Social State University of Russia Wilhelm Pieck St. 4, Moscow Russian Federation and School of Public Policy The Russian Presidential Academy of National Economy and Public Administration Prospect Vernadskogo, 84, Moscow 119571 Russian Federation Saif A. Mouhammad∗ Department of Physics Faculty of Science Taif University Taif, AL-Haweiah Kingdom of Saudi Arabia [email protected] Abstract. The problem is to obtain the most accurate upper estimate for the absolute value of the difference between the number of integer points in a circle and its area (when the radius tends to infinity). In this paper we obtain an integral equation for the function expressing the dependence of the number of integer points in a circle on its radius. The kernel of the equation contains the Bessel functions of the first kind, and the equation itself is a kind of the Hankel transform. Keywords: Gauss circle problem, integral equation, Hankel transform. 1. The problem and calculations The Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin andp with given radius. Let us consider the circle K(R): x2 + y2 ≤ R and let A( R) be the number of ∗. Corresponding author INTEGRAL EQUATION FOR THE NUMBER OF INTEGER POINTS IN A CIRCLE 523 p points with integer coordinates within this circle.
    [Show full text]
  • Laplace Transforms: Theory, Problems, and Solutions
    Laplace Transforms: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved 1 Contents 43 The Laplace Transform: Basic Definitions and Results 3 44 Further Studies of Laplace Transform 15 45 The Laplace Transform and the Method of Partial Fractions 28 46 Laplace Transforms of Periodic Functions 35 47 Convolution Integrals 45 48 The Dirac Delta Function and Impulse Response 53 49 Solving Systems of Differential Equations Using Laplace Trans- form 61 50 Solutions to Problems 68 2 43 The Laplace Transform: Basic Definitions and Results Laplace transform is yet another operational tool for solving constant coeffi- cients linear differential equations. The process of solution consists of three main steps: • The given \hard" problem is transformed into a \simple" equation. • This simple equation is solved by purely algebraic manipulations. • The solution of the simple equation is transformed back to obtain the so- lution of the given problem. In this way the Laplace transformation reduces the problem of solving a dif- ferential equation to an algebraic problem. The third step is made easier by tables, whose role is similar to that of integral tables in integration. The above procedure can be summarized by Figure 43.1 Figure 43.1 In this section we introduce the concept of Laplace transform and discuss some of its properties. The Laplace transform is defined in the following way. Let f(t) be defined for t ≥ 0: Then the Laplace transform of f; which is denoted by L[f(t)] or by F (s), is defined by the following equation Z T Z 1 L[f(t)] = F (s) = lim f(t)e−stdt = f(t)e−stdt T !1 0 0 The integral which defined a Laplace transform is an improper integral.
    [Show full text]
  • Solution to Volterra Singular Integral Equations and Non Homogenous Time Fractional Pdes
    Gen. Math. Notes, Vol. 14, No. 1, January 2013, pp. 6-20 ISSN 2219-7184; Copyright © ICSRS Publication, 2013 www.i-csrs.org Available free online at http://www.geman.in Solution to Volterra Singular Integral Equations and Non Homogenous Time Fractional PDEs A. Aghili 1 and H. Zeinali 2 1,2 Department of Applied Mathematics Faculty of Mathematical Sciences, University of Guilan, P.O. Box- 1841, Rasht – Iran 1E-mail: [email protected] 2E-mail: [email protected] (Received: 8-10-12 / Accepted: 19-11-12) Abstract In this work, the authors implemented Laplace transform method for solving certain partial fractional differential equations and Volterra singular integral equations. Constructive examples are also provided to illustrate the ideas. The result reveals that the transform method is very convenient and effective. Keywords : Non-homogeneous time fractional heat equations; Laplace transform; Volterra singular integral equations. 1 Introduction In this work, the authors used Laplace transform for solving Volterra singular integral equations and PFDEs. Solution to Volterra Singular Integral… 7 The Laplace transform is an alternative method for solving different types of PDEs. Also it is commonly used to solve electrical circuit and systems problems. In this work, the authors implemented transform method for solving the partial fractional heat equation which arise in applications. Several methods have been introduced to solve fractional differential equations, the popular Laplace transform method, [ 1 ] , [ 2 ] , [ 3 ], [ 4 ] , and operational method [ 10]. However, most of these methods are suitable for special types of fractional differential equations, mainly the linear with constant coefficients. More detailed information about some of these results can be found in a survey paper by Kilbas and Trujillo [10].
    [Show full text]
  • Math 551 Lecture Notes Fredholm Integral Equations (A Brief Introduction)
    MATH 551 LECTURE NOTES FREDHOLM INTEGRAL EQUATIONS (A BRIEF INTRODUCTION) Topics covered • Fredholm integral operators ◦ Integral equations (Volterra vs. Fredholm) ◦ Eigenfunctions for separable kernels ◦ Adjoint operator, symmetric kernels • Solution procedure (separable) ◦ Solution via eigenfunctions (first and second kind) ◦ Shortcuts: undetermined coefficients ◦ An example (separable kernel, n = 2) • Non-separable kernels (briefly) ◦ Hilbert-Schmidt theory Preface Read the Fredholm alternative notes before proceeding. This is covered in the book (Section 9.4), but the material on integral equations is not. For references on integral equa- tions (and other topics covered in the book too!), see: • Riley and Hobson, Mathematical methods for physics and engineering (this is an extensive reference, also for other topics in the course) • Guenther and Lee, Partial differential equations of mathematical physics and integral equations (more technical; not the best first reference) • J.D. Logan, Applied mathematics (more generally about applied mathematics tech- niques, with a good section on integral equations) 1. Fredholm integral equations: introduction Differential equations Lu = f are a subset of more general equations involving linear op- erators L. Here, we give a brief treatment of a generalization to integral equations. To motivate this, every ODE IVP can be written as an `integral equation' by integrating. For instance, consider the first order IVP du = f(x; u(x)); u(a) = u : (1.1) dx 0 Integrate both sides from a to x to get the integral equation Z x u(x) = u0 + f(s; u(s)) ds: (1.2) a If u solves (1.2) then it also solves (1.1); they are `equivalent' in this sense.
    [Show full text]
  • Chapter 1 Introduction to the Integral Equation(IE) and Construction Of
    Chapter 1 Introduction to the Integral Equation(IE) and Construction of the IE. 1.1. Introduction: Integral Equation began to appear since the mid-seventeenth century , when some scientists were not able to solve the differential equation . The integral equation developed with appear Abel kernel after that Volterra integral equation lastly Fredholm integral equation. In this time we find numerical method played a great role to solve integral equation. Therefore of great progressing in basic science whether physical or engineering has played essential role. Topics on integral equations has grown and evolved to its direct association lists the large branches of mathematics, such as account differential and integrative and questions of boundary conditions. During the twenty-five last year, there is a marked increase in the use of integral equations and formulations for finding scientific solutions to engineering problems and solving differential equations that are difficult to solve by normal methods. In the recent period found that the integral equations give a better solution than give differential equations. The explosive growth in industry and technology requires constructive adjustments in mathematics text researches. The integral equation it equation that appear in the unknown function under signal or more, from the signals of integrity. There more types of integral equation from it linear and nonlinear integral equation. The general formula linear integral equation it is: ١ () = () + (, )() (1 − 1) Where () unknown function, () known function and (, )known function are called kernel integral equation. We say that integral equation it is linear if that which operations on unknown function in equation it linear operations. And the general formula nonlinear integral equation it is : () = () + (, )(()) (1 − 2) Where unknown function it is nonlinear .
    [Show full text]
  • The Solution of Convolution-Typed Volterra Integral Equation by G-Transform
    International Journal of Engineering & Technology, 7 (4) (2018) 6665-6669 International Journal of Engineering & Technology Website: www.sciencepubco.com/index.php/IJET doi: 10.14419/ijet.v7i4.26348 Research paper The solution of convolution-typed Volterra integral equation by G-transform Hwajoon Kim1* and Kamsing Nonlaopon2 1Kyungdong University 2Khon Kaen University *Corresponding author E-mail:[email protected] Abstract We would like to consider the solution of convolution-typed Volterra integral equation by using G-transform, a generalized Laplace-typed transform. The tool of G-transform is analyzed to be well applied to convolution-typed Volterra integral equation. Keywords: initial value problem; G-transform; Volterra integral equation 1. Introduction It is a well-known fact that Volterra integral equation is equivalent to the initial value problem of ordinary differential equation(ODE). Since the tool used to find the solutions in this article is G-transform, a generalized Laplace-typed transform, we would like to check it up first. In a word, G-transform is a comprehensive form of Laplace-typed transform, and this transform has a strong point in the choice of an integer a. This means that we can freely choose a for a variety of problems. The form of Laplace-typed transform is Z ¥ k(s;t) f (t)dt; 0 and Laplace transform has the kernel k(s;t) = e−st as we know already[8]. This Laplace transform can be rewritten as Z ¥ − t e u f (t)dt 0 by s = 1=u, and so we proposed the general form of Laplace-typed transform by Z ¥ a − t u e u f (t)dt 0 as a natural extension[4].
    [Show full text]
  • A Technique for Solving the Singular Integral Equations of Potential Theory
    Old Dominion University ODU Digital Commons Mathematics & Statistics Theses & Dissertations Mathematics & Statistics Spring 2007 A Technique for Solving the Singular Integral Equations of Potential Theory Brian George Burns Old Dominion University Follow this and additional works at: https://digitalcommons.odu.edu/mathstat_etds Part of the Mathematics Commons Recommended Citation Burns, Brian G.. "A Technique for Solving the Singular Integral Equations of Potential Theory" (2007). Doctor of Philosophy (PhD), Dissertation, Mathematics & Statistics, Old Dominion University, DOI: 10.25777/q9kv-0x34 https://digitalcommons.odu.edu/mathstat_etds/6 This Dissertation is brought to you for free and open access by the Mathematics & Statistics at ODU Digital Commons. It has been accepted for inclusion in Mathematics & Statistics Theses & Dissertations by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. A TECHNIQUE FOR SOLVING THE SINGULAR INTEGRAL EQUATIONS OF POTENTIAL THEORY by Brian George Burns M.Sci. 1998, University of Glasgow M.Sc. 1999, University of Stirling M.S. 2005, Old Dominion University A Dissertation Submitted to the Faculty of Old Dominion University in Partial Fulfillment of the Requirement for the Degree of DOCTOR OF PHILOSOPHY COMPUTATIONAL AND APPLIED MATHEMATICS OLD DOMINION UNIVERSITY May 2007 John Tweed (Director) Ion Melrose John/A. Adam Richard D. Noren Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT A TECHNIQUE FOR SOLVING THE SINGULAR INTEGRAL EQUATIONS OF POTENTIAL THEORY Brian George Burns Old Dominion University, 2007 Director: Dr. John Tweed The singular integral equations of Potential Theory are investigated using ideas from both classical and contemporary mathematics. The goal of this semi-analytic ap­ proach is to produce numerical schemes that are both general and computationally simple.
    [Show full text]
  • Numerical Solutions of Fredholm Integral Equation of Second Kind Using Piecewise Bernoulli Polynomials
    Numerical Solutions of Fredholm Integral Equation of Second Kind Using Piecewise Bernoulli Polynomials Afroza Shirin1, Md. Shafiqul Islam*2 1Institute of Natural Sciences, United International University, Dhaka-1209, Bangladesh. Email: [email protected]. 2Department of Mathematics, University of Dhaka, Dhaka – 1000, Bangladesh *Corresponding author, Email: [email protected] Abstract: The aim of this paper is to solve the integral equations numerically using piecewise Bernoulli polynomials. The Bernoulli polynomials are derived explicitly over the unit interval. A matrix formulation for a non-singular linear Fredholm integral equation of the second kind is derived by the technique of Galerkin method. In the Galerkin method, the Bernoulli polynomials are exploited as the linear combination in the approximation as basis functions. Numerical examples are considered to verify the effectiveness of the proposed derivations. Keywords: Fredholm integral equation, Galerkin method, Bernoulli polynomials, Numerical solutions. I. Introduction In the survey of solutions of integral equations, a large number of analytical but a few approximate methods are available for solving numerically various classes of integral equations [1, 2, 7, 8 ]. Since the piecewise polynomials are differentiable and integrable, the Bernstein polynomials [5 – 8] have been used for solving differential and integral equations numerically. Recently, integral equations have been solved by the well known variational iteration method [9]. In the literature [7], Mandal and Bhattacharya have attempted to solve integral equations numerically using Bernstein polynomials, but they obtained the results in terms of finite series solutions In contrast to this, we solve the linear Fredholm integral equation of the second kind by exploiting very well known Galerkin method, and Bernoulli polynomials [4] are used as trial functions.
    [Show full text]
  • Integral Equation
    :Integral Equation: 1. Definition Integral Equation: An integral equation is an equation is which an unknown function appears under one or more integral sign. 2. Linear and Non-Linear Integral equation: An integral equation is called linear if only linear operation are performed in it upon the unknown function. An integral equation which is not linear is known as a non-linear integral equation. Most general type of integral equation is of the form ( ) ( ) ( ) ( ) ( ) . In this case upper limit may be variable or constant. And = + ∫ , lower limit is always constant. 3. If upper limit is constant then integral equation is known as Fredholm integral equation. If upper limit is variable ( ) then integral equation is known as Volterra integral equation. 4. In above expression if () ≠ then integral equation is third kind. And if () = 0 then integral equation is first kind. Again when () = 1 then integral equation is second kind. If () = 0 then integral equation is homogeneous. 5. Singular Integral equation: When one or both limits of integration become infinite or when the kernel become infinite at one or more points within the range of integration, the integral equation is known as singular integral equation. 6. Symmetric kernel: A kernel (, ) is symmetric (hermitian) if (, ) = (, ) in case of real values = . 7. Seprable of Degenerate kernel: (, ) = ∑ () ℎ() note that () can be regarded as linearly independent. 8. Integral equation of the Convolution type: Kernel must be of type (, ) = ( − ) 9. Iterated Kernels: Iterated kernels (, ), = 1,2,3, … are defined as: (, ) = (, ) and ( ) ( ) ( ) , = ∫ , , = 2,3, …. 10. Resolvent Kernel or Reciprocal Kernel: It written in place of (, ) to find solution in form (, ; ) Γ(x, t; λ) 11.
    [Show full text]
  • SOLUTION of INTEGRAL EQUATIONS and LAPLACE - STIELTJES TRANSFORM Deshna Loonker
    Palestine Journal of Mathematics Vol. 5(1) (2016) , 43–49 © Palestine Polytechnic University-PPU 2016 SOLUTION OF INTEGRAL EQUATIONS AND LAPLACE - STIELTJES TRANSFORM Deshna Loonker Communicated by P. K. Banerji MSC 2010 Classifications: Primary 44A35; Secondary 44A99, 45D05, 45E10, 46F10, 46F12 Keywords : Laplace - Stieltjes transform, Laplace transform, distribution spaces, Volterra integral equation, Fredlom integral equation of convolution type, convolution. This work is partially supported by the UGC Post Doctoral Fellowship for Women, India, No. F.15-134/12 (SA-II), sanctioned to the author. Abstract We have obtained solutions of integral equations such as Volterra convolution type of first and second type and Abel integral equation using Laplace - Stieltjes transform and ap- plications are mentioned through examples. The convolution property for Laplace - Stieltjes transform is obtained. The solution obtained is considered in distributional sense. 1 Introduction Integral transforms, viz. Fourier, Laplace, Hankel, Sumudu, Elzaki, Aboodh, and the wavelet transform among other, have been used [1, 3, 4, 5, 6, 7, 8] to obtain solutions of various types of differential and integral equations. The Laplace transform can be studied and researched from years ago [1, 9] In this paper, Laplace - Stieltjes transform is employed in evaluating solutions of certain integral equations that is aided by the convolution. It is remarked that the solution of integral equations obtained by using Laplace - Stieltjes transform, can also be defined on distribution spaces. Let F (t) is a well defined function of t, for t ≥ 0 and s be a complex number. The Stieltjes integral is 1 Z ∗ −st (LsF )(s) = F (s) = e dF (t); (1.1) 0 which converges on some s0; the Stieltjes integral (1.1) converges on s such that Re(s) > Re(s0).
    [Show full text]
  • Direct Method to Solve Volterra Integral Equation of the First Kind
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 220 (2008) 51–57 www.elsevier.com/locate/cam Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions E. Baboliana,∗, Z. Masourib aDepartment of Mathematics, Teacher Training University, Tehran, Iran bDepartment of Mathematics, Islamic Azad University, Science and Research Campus, Tehran, Iran Received 26 April 2007; received in revised form 16 July 2007 Abstract This article proposes a simple efficient direct method for solving Volterra integral equation of the first kind. By using block-pulse functions and their operational matrix of integration, first kind integral equation can be reduced to a linear lower triangular system which can be directly solved by forward substitution. Some examples are presented to illustrate efficiency and accuracy of the proposed method. © 2007 Elsevier B.V. All rights reserved. MSC: 65R20; 45A05; 45D05; 41A30 Keywords: Volterra integral equation of the first kind; Direct method; Vector forms; Block-pulse functions; Operational matrix 1. Introduction The integral equation method is widely used for solving many problems in mathematical physics and engineering. These problems are often reduced to integral equations of the first kind. This article proposes a novel direct method to solve these equations. Integral equations of the first kind are inherently ill-posed problems, meaning that the solution is generally unstable, and small changes to the problem can make very large changes to the answers obtained [2,4,5]. This ill-posedness makes numerical solutions very difficult, a small error may lead to an unbounded error.
    [Show full text]